Force and Motion

Learning Goals:

-Differentiate between scalars (distance, speed) and vectors (displacement, velocity)-Explain the relationship between acceleration and velocity-Define g, the acceleration due to gravity-Describe and understand centripetal acceleration-Understand projectile motion -Understand and apply the three laws of Newton and the law of Universal gravitation-Explain the conditions for the conservations of linear and angular momenta

Motion

• To describe completely the motion of an object you need to know three things– Position (at one single time)

• Cartesian coordinates• Needs a reference point

– Velocity • Speed (scalar) AND direction → VECTOR

– Acceleration• Magnitude (scalar) AND direction → VECTOR

Scalar VS Vector

• Scalar: magnitude (and units, of course!)– Ex.: your mass, the temperature, SPEED,

distance, etc.

• Vector: magnitude (and units) AND DIRECTION!– Ex.: velocity, force, displacement, etc.– Are represented by arrows

• Length proportional to the magnitude• Arrowhead indicates the direction

Moving from place to place

• Velocity: VECTOR– Average velocity = displacement / time to travel

• Displacement: straight line distance between starting and ending point, with direction pointing towards the end point.

• Ex.: 2h to go to L.A. (180km), 4h to come backAverage velocity = 0[km] / (2[h]+ 4[h]) = 0 [km/h] !!!

– Instantaneous velocity = speedometer reading + direction!!!

t

dv

• Speed: SCALAR– Average speed = Distance Traveled/time to travel OR

• Ex.: 2h to go to L.A. (180km), 4h to come backAverage Speed = (180[km]+180[km])/(2[h] + 4[h]) = 60 [km/h]

– Instantaneous Speed: speedometer reading

Displacement is a vector quantity between two points.Distance is the actual path traveled.

Special Case: Constant Velocity

• If the velocity is constant then– The speed is not changing– The direction is not changing (i.e. straight line motion!)

• We choose one direction to be +, the other to be –

Ex. At a track meet, a runner runs the 100.m dash in 15s. What was the runner’s average speed? Average velocity?

[m/s]7.615[s]

]m.[100

t

dv

Acceleration

• If the velocity changes, there is acceleration– Change in MAGNITUDE and/or the DIRECTION – Ex. : car slowing down or speeding up, the Earth around the

Sun, a ball falling down

• Average acceleration = change in velocity/ time for change

=

vo = starting velocity

vf = ending velocity

The UNITS of acceleration are (m/s)/s = m/s2

t

vv

t

va of

Acceleration (cont’d)• Acceleration is a vector

– Magnitude: rate at which the velocity is changing– Direction: how is the velocity changing

• If velocity and acceleration are in the same direction, the speed INCREASES

• If velocity and acceleration are in opposite direction, the speed DECREASES

Which of these car is accelerating?

A. A car on a circular race track going at a constant speed

B. A car coasting to a stop

C. A and B

D. None of these car is accelerating.

Special Case: gravitational acceleration g

• Objects falling at the surface of the Earth ALL fall with the same acceleration g=9.80m/s2, down

• Without the effect of AIR resistance = free fall

• Dropping: Distance fallen =

• t = time for falling

2

2

1gtd

A ball is dropped from a tall building. How far does the ball drop in 0.50 s?

2

2

1gtd

22 ]s[50.0]m/s[80.92

1d

]m[2.1d

Throwing a ball up, letting it fall down…

• There is gravitational acceleration (ALWAYS DOWN) when the ball goes up as well as when it comes down– Way Up: acceleration

opposite velocity so SLOWING DOWN

– Way Down: acceleration same direction as velocity so SPEEDING UP

Uniform Circular Motion

• Constant speed BUT direction is changing all the time!– There is an ACCELERATION!!!– Points towards the center of the circular path

– Called Centripetal Acceleration ac

– ac=

– v = speed, r = radius of the circular path

r

v2

The string keeps the ball in circular motion.

If you cut the string, the ball will continue in a straight line.

Uniform Circular Motion (cont’d)

Ex.: A person drives a car around a circular racetrack with a radius of 70.m at 10.m/s. What is the acceleration of the car?

r

vac

2

]m/s[4.1]m.[70

])m/s.[10( 22

ca

Projectile Motion

• Any object thrown by some means– Golf ball, tennis ball, football, bullet, etc.

• The HORIZONTAL (parallel to the ground) and VERTICAL (perpendicular to the ground) motions are INDEPENDANT

If there were no gravity

Gravity with high speed

Gravity with slow speed Aiming above the monkey’s head

A zookeeper wants to feed a banana to the monkey with his cannon. The monkey always let go of the branch when the banana is shot. Where should he aim: above, below or on???

Forces and Net Force

• Forces are CAPABLE of producing a change in velocity.– ONLY if the force is unbalanced– Forces are VECTORS

• If there is a net, or unbalanced, force: the motion will CHANGE!– There will be an acceleration

Laws of Newton

1. An object will remain at rest or in uniform motion in a straight line unless acted on by an external, unbalanced force.

• The greater the mass of an object, the harder it is to change its motion : INERTIA

• An object doesn’t need a force to keep on moving!!!

• Examples of forces? • Apart from gravity, in your daily lives, force is transmitted

through contact

An elevator is being lifted up an elevator shaft at a constant speed by a steel cable. Forces on the elevator are such that:

A. The upward force by the cable is greater than the downward force of gravity

B. The upward force by the cable is equal to the downward force of gravity

C. The upward force by the cable is smaller than the downward force of gravity

D. None of the above.

An object keeps on moving the way it was UNLESS a force acts on it…

Explain those Examples using Newton’s first law:

1. Blood rushes from your head to your feet while quickly stopping when riding on a descending elevator.

2. The head of a hammer can be tightened onto the wooden handle by banging the bottom of the handle against a hard surface.

3. To dislodge ketchup from the bottom of a ketchup bottle, it is often turned upside down and thrusted downward at high speeds and then abruptly halted.

4. Headrests are placed in cars to prevent whiplash injuries during rear-end collisions.

5. While riding a skateboard (or wagon or bicycle), you fly forward off the board when hitting a curb or rock or other object which abruptly halts the motion of the skateboard.

2. Fnet = ma

• Unbalanced Force = mass × acceleration

OR

mass

force Unbalancedonaccelerati

• For the same force acting on difference object, the heavier the object, the smaller the acceleration.

• For a given object, the larger the force acting on it, the larger the acceleration.

• Different objects will have the same acceleration if a force proportional to their mass is applied on them…

A net external force of 21 N is applied to a mass of 3.0 kg. From Newton’s 2nd law, what will be the resulting acceleration?

A. 63 m/s2

B. 21 m/s2

C. 7.0 m/s2

D. Zero.

Centripetal Acceleration: caused by a FORCE!

3. For every action there is an equal and opposite reaction.

• Force always come in pair, acting between two different object!!!— The Earth attracts you towards it by gravity, you attract

it towards YOU with the same force.— When a train and a Beetle collide, the train AND the

Beetle will have the same force hitting them.— When you walk, you push on the ground BACK and

the ground reacts by pushing you FORWARD with the same force.

— Recoil from a fired riffle: the bullet is pushed forward with the same force that pushes back on the riffle.

— If you are stranded in space with nothing… you are dooooomed!

A large truck breaks down out on the road and receives a push back into town by a small compact car.

1. While the car, still pushing the truck, is speeding up to get up to cruising speed:

A. The amount of force with which the car pushes on the truck is equal to that with which the truck pushes back on the car

B. The amount of force with which the car pushes on the truck is smaller than that with which the truck pushes back on the car.

C. The amount of force with which the car pushes on the truck is greater than that with which the truck pushes back on the car

D. Neither the car nor the truck exert any force on the other.

A large truck breaks down out on the road and receives a push back into town by a small compact car.

2. After the car reaches the constant cruising speed at which its driver wishes to push the truck:

A. The amount of force with which the car pushes on the truck is equal to that with which the truck pushes back on the car.

B. The amount of force with which the car pushes on the truck is smaller than that with which the truck pushes back on the car.

C. The amount of force with which the car pushes on the truck is greater than that with which the truck pushes back on the car.

D. Neither the car nor the truck exert any force on the other.

Newton’s Law of Gravitation

• FG = Force of gravity

• G = 6.67x10-11 Nm2/kg2

• m1 and m2: the TWO masses that are attracting each other

• r: the distance between their centers

221

r

mGmFG

What would the force of gravity be between:

1. the Earth and an apple of 0.25kg on a table?

2. the Earth and a football player of 85kg on a football field?

3. the Earth and a 1250kg elephant at the Wild Animal Park?

4. The Earth and a 85kg astronaut in the Space Station 400km above ground?

?22?

221

s

m81.9 m

R

mmG

r

mGmF

E

EG

2.5N

830N

12300N

740N

Linear and Angular momenta

• Linear momentum p = mv → vector!– m: mass, v: velocity

• Angular momentum L = mvr → vector!– m: mass, v: velocity, r: separation with center

These two quantities will remain constant for a group of objects UNLESS an EXTERNAL UNBALANCED force is applied to them.

Linear momentum• Pi = Pf = 0 (for man and boat)• When the man jumps out of the

boat he has momentum in one direction and, therefore, so does the boat, but in the opposite direction.

• Their momenta must cancel out! (= 0)

Angular Momentum • force that can create rotation: force that can generate torque• If there are no torques acting on an object, the angular

momentum is conserved (L doesn’t change).

L = mrv

• Planets are in elliptical motion around the Sun• Sometimes they are closer (r gets smaller) and

sometimes they are further (r gets larger) from the Sun.

L = mrv

Homework

• Chapter 2– Short-answer questions

• 4, 7, 10, 18

– Exercises• 2

• Chapter 3– Short-answer questions

• 1, 8, 9, 20

– Exercises• 2, 4